12. Static Equilibrium

1. Conditions for Equilibrium

2. Center of Gravity

3. Examples of Static Equilibrium

4. Stability

The Alamillo Bridge in Seville, is the work of architect Santiago Calatrava.

What conditions must be met to ensure the stability of this dramatic?

= =net netF τ 0

12.1. Conditions for Equilibrium

(Mechanical) equilibrium = zero net external force & torque.

Static equilibrium = equilibrium + at rest.

=iF 0

=i τ 0 i i r F

Pivot point = origin of ri .

is the same for all choices of pivot points=iF 0 i τ

Prob 55:

For all pivot points

Example 12.1. Drawbridge

The raised span has a mass of 11,000 kg uniformly distributed over a length of 14 m.

Find the tension in the supporting cable.

=i τ 0

Force Fh at hinge not known.

Choose pivot point at hinge.

g T τ τ 0

1 2sin sin 02

Lm g L T

1 90 30 120

2 180 30 15 165

1

2

sin

2sin

m gT

211,000 9.8 / sin120

2sin165

kg m s

180 kN Another choice of pivot: Ex 15

y

x30

Tension T

Gravity mgHinge force Fh

15

2

1

GOT IT? 12.1.

Which pair, acting as the only forces on the object, results in static equilibrium?

Explain why the others don’t.

(C)

(A): F 0.

(B): 0.

12.2. Center of Gravity

i i τ r F i im r g

Total torque on mass M in uniform gravitational field :

= i im r g

cm M τ r g

Center of gravity = point at which gravity seems to act

cg cmr r for uniform gravitational field

net cg net τ r F

CG does not exist if net is not Fnet .

Conceptual Example 12.1. Finding the Center of Gravity

1st pivot

2nd pivot

Explain how you can find an object’s center of gravity by suspending it from a string.

GOT IT? 12.2.

The dancer in the figure is balanced; that is, she’s in static equilibrium.

Which of the three lettered points could be her center of gravity?

12.3. Examples of Static Equilibrium

All forces co-planar: =iF 0

=i τ 0

2 eqs in x-y plane

1 eq along z-axis

Tips: choose pivot point wisely.

Example 12.2. Ladder Safety

A ladder of mass m & length L leans against a frictionless wall.

The coefficient of static friction between ladder & floor is .

Find the minimum angle at which the ladder can lean without slipping.

Fnet x : 1 2 0n n

Fnet y : 1 0n m g

Choose pivot point at bottom of ladder.

z : 2 sin 180 sin 90 02

LL n m g

2 1n nm g

2 sin cos 02

LL n m g

2

tan2

m g

n

1

2

0 90

y

x

mgn1

fS = n1i

n2

Example 12.3. Arm Holding Pumpkin

Find the magnitudes of the biceps tension & the contact force at the elbow joint.

Fnet x : cos 0c xF T

Fnet y : sin 0c yT F m g M g

Pivot point at elbow.

z : 1 2 3sin 0x T x m g x M g

2 3

1 sin

x m x M gT

x

20.036 2.7 0.32 4.5 9.8 /

0.036 sin80

m kg m kg m s

m

500 N

cosc xF T

sinc yF T m M g

500 cos80N 87 N

2500 sin80 2.7 4.5 9.8 /N kg kg m s 420 N

2 2c c x c yF F F 430 N ~ 10 M g

y

x

Mgmg

T

Fc80

GOT IT? 12.3.

A person is in static equilibrium leaning against a wall.

Which of the following must be true:

(a) There must be a frictional force at the wall but not necessarily at the floor.

(b) There must be a frictional force at the floor but not necessarily at the wall.

(c) There must be frictional forces at both floor and wall.

Need frictional force to balance normal force from wall.

Application: Statue of Liberty

Sculptor Bartholdi : lasting as long as the pyramids.

Deviation from Eiffel’s plan resulted in excessive torque.

Major renovation was required after only 100 yrs.

12.4. Stability

Stable equilibrium: Original configuration regained after small disturbance.

Unstable equilibrium: Original configuration lost after small disturbance.

Stable equilibrium

unstable equilibrium

Stable

Unstable

Neutrally stable

Metastable

Equilibrium: Fnet = 0.

V at global min

V at local max

V = const

V at local min

2

20

d V

d x

2

20

d V

d x

2

20

d V

d x

2

20

d V

d x

0d V

d x

Stable equilibrium : PE at global min

Metastable equilibrium : PE at local min

Example 12.4. Semiconductor Engineering

A new semiconductor device has electron in a potential U(x) = a x2 – b x4 ,

where x is in nm, U in aJ (1018 J), a = 8 aJ / nm2, b = 1 aJ / nm4.

Find the equilibrium positions for the electron and describe their stability.

Equilibrium criterion : 0d U

d x

32 4 0a x b x

2 nm

0x

2

ax

bor

2

4

8 /

2 1 /

aJ nm

aJ nm

22

22 12

d Ua b x

d x

2

2

0

2 0x

d Ua

d x

x = 0 is (meta) stable

2

2

/2

4 0x a b

d Ua

d x

x = (a/2b) are unstable

equilibria

Metastable

Saddle Point

, ,0

U x y U x y

x y

Equilibrium condition

2

2

,0

U x y

x

Saddle point

stable

2

2

,0

U x y

y

unstable

stable

unstable

GOT IT? 12.4

Which of the labeled points are stable, metastable, unstable, or neutrally stable equilibria?

SM

UU

N